On Lappan's five-point theorem
type: Item
instance of: scholarly article
title: On Lappan's five-point theorem
zbMATH Open document ID: 1370.30018
author: Tran Van Tan
author: Nguyen Van Thin
published in: Computational Methods and Function Theory
publication date: 2017-04-21T00:00:00Z
review text: \textit{A. Hinkkanen} [N. Z. J. Math. 22, No. 2, 39--41 (1993; Zbl 0796.30029)] and \textit{P. Lappan} [Rev. Roum. Math. Pures Appl. 39, No. 7, 691--702 (1994; Zbl 0843.30031)] proved that a family \(\mathfrak{F}\) of meromorphic functions in a domain \(D \subset \mathbb{C}\) is normal if and only if for each compact set \(K \subset D\), there exist a set \(E = E(K) \subset \widehat{\mathbb{C}}\) containing at least five distinct points and a positive constant \(M = M(K)\) such that \(\displaystyle \max_{z \in K \cap f^{-1}(E)} f^{\#}(z) \leq M\) for all \(f \in \mathfrak{F}\). In the paper under review the authors prove an analogue of the `if' part of the above result for a set \(E = E(K)\) containing four and three distinct points. In addition to the spherical derivative of the members of \(\mathfrak{F}\) their results involve also the sperical derivative of the first and the second derivatives. They further prove similar results for \(\phi\)-normal functions.
reviewed by: Indrajit Lahiri
Mathematics Subject Classification ID: 30D45
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 6707035
zbMATH Keywords: normal family
zbMATH Keywords: normal function
zbMATH Keywords: Nevanlinna theory
MaRDI profile type: Publication
full work available at URL: s40315-016-0168-9
OpenAlex ID: W2461921294
cites work: Properties of meromorphic \(\varphi \)-normal functions
cites work: Q4867380
cites work: The spherical derivative of integral and meromorphic functions
cites work: Normal criteria for families of meromorphic functions
cites work: Q5723337
cites work: On a result of Chuang related to Hayman's alternative
cites work: Q4291018
cites work: A criterion for a meromorphic function to be normal
cites work: Boundary behaviour and normal meromorphic functions
cites work: Q4526388
cites work: Problems in Complex Function Theory
cites work: Normal families: New perspectives
DOI: 10.1007/S40315-016-0168-9
description: scientific article; zbMATH DE number 6707035